When `((1)/(2) - (1)/(4) + (1)/(5) - (1)/(6))` is divided by `((2)/(5) - (5)/(9) + (3)/(5) - (7)/(18))`, the result is :

Options

A. `5 (1)/(10)`
B. `2 (1)/(18)`
C. `3 (1)/(6)`
D. `3 (3)/(10)`

Answer

To solve the problem of dividing the expression \(\left(\frac{1}{2} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6}\right)\) by \(\left(\frac{2}{5} - \frac{5}{9} + \frac{3}{5} - \frac{7}{18}\right)\), we will follow these steps:

### Step 1: Simplify the numerator
We start with the expression in the numerator:
\[ \frac{1}{2} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} \]
To simplify this, we first find a common denominator. The least common multiple (LCM) of \(2\), \(4\), \(5\), and \(6\) is \(60\).

Now, we convert each fraction:
\[ \frac{1}{2} = \frac{30}{60}, \quad \frac{1}{4} = \frac{15}{60}, \quad \frac{1}{5} = \frac{12}{60}, \quad \frac{1}{6} = \frac{10}{60} \]
Substituting these values into the expression gives:
\[ \frac{30}{60} - \frac{15}{60} + \frac{12}{60} - \frac{10}{60} \]
Now, we combine these fractions:
\[ \frac{30 - 15 + 12 - 10}{60} = \frac{17}{60} \]

### Step 2: Simplify the denominator
Next, we simplify the expression in the denominator:
\[ \frac{2}{5} - \frac{5}{9} + \frac{3}{5} - \frac{7}{18} \]
Again, we find a common denominator. The LCM of \(5\), \(9\), and \(18\) is \(90\).

Now, we convert each fraction:
\[ \frac{2}{5} = \frac{36}{90}, \quad \frac{5}{9} = \frac{50}{90}, \quad \frac{3}{5} = \frac{54}{90}, \quad \frac{7}{18} = \frac{35}{90} \]
Substituting these values into the expression gives:
\[ \frac{36}{90} - \frac{50}{90} + \frac{54}{90} - \frac{35}{90} \]
Now, we combine these fractions:
\[ \frac{36 - 50 + 54 - 35}{90} = \frac{5}{90} = \frac{1}{18} \]

### Step 3: Divide the simplified numerator by the simplified denominator
Now we divide the results from Step 1 and Step 2:
\[ \frac{\frac{17}{60}}{\frac{1}{18}} = \frac{17}{60} \times \frac{18}{1} = \frac{17 \times 18}{60} = \frac{306}{60} \]
Now, we simplify \(\frac{306}{60}\):
\[ \frac{306 \div 6}{60 \div 6} = \frac{51}{10} \]

### Step 4: Convert to mixed number
Finally, we convert \(\frac{51}{10}\) into a mixed number:
\[ 51 \div 10 = 5 \quad \text{remainder } 1 \]
Thus, \(\frac{51}{10} = 5 \frac{1}{10}\).

### Final Result
The final result is:
\[ 5 \frac{1}{10} \]